generic model control for lithium-ion batteriesdepts.washington.edu/maple/pubs/61_j....

Journal of The Electrochemical Society, 164 (6) A973-A986 (2017) A973

Generic Model Control for Lithium-Ion BatteriesManan Pathak,a,∗ Suryanarayana Kolluri,a,∗ and Venkat R. Subramaniana,b,∗∗,z

aDepartment of Chemical Engineering, University of Washington, Seattle, Washington98195, USAbPacific Northwest National Laboratory, Richland, Washington 99352, USA

Battery Management Systems (BMS) are critical to safe and efficient operation of lithium-ion batteries and accurate predic-tion of the internal states. Smarter BMS that can estimate and implement optimal charging profiles in real-time are impor-tant for advancement of the Li-ion battery technology. Estimating optimal profiles using physics-based models is computa-tionally expensive because of the non-linear and stiff nature of the model equations, involving the need for constrained non-linear optimization. In this work, we present an alternative approach to control batteries called as Generic Model Control,or Reference System Synthesis. This work enables robust stabilization and control of battery models to set-point as an al-ternative approach, eliminating the need to perform optimization of nonlinear models. As compared to the generic modelcontrol approaches implemented by previous researchers, we implement the same concept using direct DAE numericalsolvers. The results are presented for single input single objective problems, and for constrained problems for various batterymodels.© The Author(s) 2017. Published by ECS. This is an open access article distributed under the terms of the Creative CommonsAttribution Non-Commercial No Derivatives 4.0 License (CC BY-NC-ND, http://creativecommons.org/licenses/by-nc-nd/4.0/),which permits non-commercial reuse, distribution, and reproduction in any medium, provided the original work is not changed in anyway and is properly cited. For permission for commercial reuse, please email: [email protected]. [DOI: 10.1149/2.1521704jes]All rights reserved.

Manuscript submitted October 11, 2016; revised manuscript received February 16, 2017. Published March 14, 2017. This was Paper208 presented at the San Diego, California, Meeting of the Society, May 29- June 2, 2016.

Lithium-ion batteries are used in a wide range of applications rang-ing from cell phones to electric vehicles (EVs) to electric grids. Withthe expected decline in the price of batteries over the next few years,the market for EVs and grids is growing rapidly. Long rechargingtimes, capacity fade and thermal runaway are some of the main issuesthat need to be addressed in order to use batteries safely and efficientlyfor a longer time. This necessitates the use of smarter battery man-agement systems that can derive optimal use strategies by exploitingthe dynamics of the battery.

Battery models based on transport, physical, electrochemical andthermodynamics principles can be used to monitor the internal statesof the battery and to obtain optimal control strategies. Such mod-els are computationally expensive which limits their use in controlapplications in real-time. Various issues, such as the onset of dy-namics of some internal states only during use of batteries at highcharge/discharge rates, and several other states which are active onlywhile charging or discharging, adds to the challenges of the observ-ability of these physics-based electrochemical models. Hence, variousapproximated and reduced order models have been proposed in thepast, to make the models strongly observable in local intervals.1,2–5

These models are also highly non-linear in nature, which adds signifi-cant challenges to implement control strategies along with the above-mentioned issues. Past researchers have shown multivariable controltechniques like Dynamic Matrix Control (DMC),6 Internal ModelControl (IMC),7 and Model Predictive Heuristic Control (MPHC)8

for a variety of systems. However, these models suffer from their re-liability on the linear approximations of the experimentally obtainedstep-response data and do not directly consider the full nonlinearmodel explicitly.

Several researchers have been working to design optimal charg-ing profiles for batteries by implementing a wide range of controlstrategies. Methekar et al.9 derived optimal charging profiles for max-imizing the energy using the Control Vector Parameterization (CVP)approach. Rahimian et al.10 calculated the charging current to mini-mize the capacity fade vs. cycle number using a single particle model(SPM). Perez et al.11 obtained optimal charging profiles for batterieswith constraints on temperature as well as concentration of lithiumin solid and electrolyte phase. Suthar et al.12 proposed optimal charg-

∗Electrochemical Society Student Member.∗∗Electrochemical Society Member.

zE-mail: [email protected]

ing profiles for minimizing the intercalation induced stresses insideparticle using simultaneous discretization approach. Hoke et al.13 pro-posed a method to minimize the cost of battery charging in marketswith variable electricity costs after accounting for battery degradation.However, all of these methods require nonlinear and robust optimiza-tion approaches.

In this article, we present an alternative approach to control bat-teries, applying the method of Generic Model Control (GMC),14 alsoknown as Reference System Synthesis.15 This approach is based ona set-point strategy for the measured (state) variable. GMC has pre-viously been implemented on various applications such as multistageflash desalination,16 distillation column control,17–19 control of nutri-ents – removing activated sludge systems,20 armature current controlof DC motor,21 etc. In this article, we apply the GMC technique tophysics-based battery models. Lee and Sullivan14 originally proposedthe GMC algorithm for ordinary differential equations (ODEs) and forproblems in which the measured variable is not explicitly dependent onthe manipulated variable. Here, we extend the GMC technique to dif-ferential algebraic equations (DAEs) with set points for algebraic vari-ables, and for problems in which the output is directly dependent oncontrol, for applying it to battery models. Application of GMC requiresrobust DAE solvers. By combining GMC approach with robust DAEsolvers, we show the ability to reformulate some of the common con-trol and optimization problems for batteries as direct simulation prob-lems. By rewriting charging and optimization objective as set-pointcontrol problems, we derive control profiles using the GMC approach.While the theoretical internal stability for the GMC approach is yet tobe proven unlike nonlinear model predictive control approaches,22 inour experience, the GMC approach is computationally competitive asit avoids the need for optimization for unconstrained problems, andfor problems with bounds on the manipulated variable. The gain inCPU time is obtained only if efficient and robust DAE solvers areused.

In subsequent sections, we discuss the theory behind the GMCframework, followed by case studies for applying GMC control for-mulation for a thin film nickel hydroxide electrode, a single particlemodel, along with discussing various objectives for a physics-basedLi-ion battery model. For the physics-based Li-ion battery model, thefirst case demonstrates the problem formulation and implementationfor obtaining a current profile for a set-point given on the charge storedin the battery. The second case considers a set-point of 4.2 V for volt-age of the cell, without any bounds on the applied current, followedby current containing bounds.

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A974 Journal of The Electrochemical Society, 164 (6) A973-A986 (2017)

Generic Model Control

The generic model control (GMC) strategy is based on finding val-ues for the manipulated variable that forces the controlled (measured)variable to reach its set-point in a non-linear process model. The de-sired set-point and its measured variable is compared to generate theerror. The obtained error is used in GMC architecture to generate con-trol input for the system (process dynamics) to direct the controlledvariable toward the desired set-point.

The strategy for implementing generic model control is describedas follows. Consider a dynamic process model given by a set ofdifferential equations:

dx

dt= f (x, u, t) [1]

where x is the vector of state variables of dimension n, u is the scalarprocess input (also called as manipulated variable) of dimension m, tis the time. The relation for output variables is given by:

Y = M(x, u) [2]

where Y is a vector of output variables (measured variables) of dimen-sion p.

To obtain control law from above Equations 1 and 2, a referencetrajectory proposed by Lee and Sullivan14 is given as,(

dY

dt

)= K (Y set − Y ) + K

τi

∫ t

0

(Y set − Y

)dτ [3]

where Y set is the desired set-point, K and τi are the tuning parameters.The first term of the right hand side of Equation 3 is the rate of changeof Y moving toward the desired set-point Y set , and can be referred toas the Proportional term. The second term is added to minimize theoffset between the actual value and the desired set-point, and can bereferred to as the Integral term. Using Equation 3, a generic controllaw is obtained for GMC, which is given as,

dY

dt− K

(Y set − Y

) − K

τi

t∫0

(Y set − Y

)dτ = 0 [4]

The term ( dYdt ) is the reference trajectory that can be obtained using

Equations 1 and 2.(dY

dt

)= ∂ M

∂xf (x, u, t) + ∂ M

∂u· du

dt[5]

Combining Equations 4 and 5 gives the control law of GMC in termsof the process model and the reference trajectory.

∂ M

∂xf (x, u, t) + ∂ M

∂u

du

dt− K

(Y set − Y

) − K

τi

t∫0

(Y set − Y

)dτ = 0

[6]Most physics-based battery models are given by partial differentialequations (PDEs), which are converted to DAEs for simulation oroptimization. Often times, the output or the measured variable is adirect function of the manipulated variable. Below, we discuss anapproach to apply GMC for DAEs, including cases for which setpoint is given for an algebraic variable.

Consider an index-1DAE system governing two scalars y and zgiven by:

dy

dt= f (y, z, u) [7]

g(y, z, u) = 0 [8]

where, y is the differential variable, z is the algebraic variable and uis the manipulated variable. If the controlled (measured) variable isy,then Equations 4 and 6 can be used directly, as the derivative is explic-itly available from the model equations. If the measured variable is z(for example potential as in the case of batteries), then the derivative

( dzdt ) is not available directly from the model equations. The derivative

( dzdt ) in such cases, can be obtained by differentiating Equation 8 with

time to get:

dg

dt=∂g

∂y(y, z, u)· f (y, z, u)+ ∂g

∂z(y, z, u)· dz

dt+ ∂g

∂u(y, z, u)· du

dt=0

[9]GMC equation for this case is given by:(

dz

dt

)= K (zset − z) + K

τi

∫ t

0

(zset − z

)dτ [10]

Equation 10 can now be substituted in Equation 9 to get:

dg

dt= ∂g

∂y(y, z, u) · f (y, z, u) + ∂g

∂z(y, z, u) ·

[K (zset − z)

+ K

τi

∫ t

0

(zset − z

)dτ

]+ ∂g

∂u(y, z, u) · du

dt= 0 [11]

In this case, Equation 7 governs the dynamics of the variable y, whilevariables z and u are governed by Equations 8 and 11 respectively.Even though a du

dt term appears in 11, since the experimental value ofz is known initially at time t = 0, the initial condition for u can becalculated using 8. The final form of the equations to be solved forobtaining the control profile is summarized in 12 as:

dy

dt= f (y, z, u) (a)

g(y, z, u) = 0 (b)

dg

dt= ∂g

∂y(y, z, u) · f (y, z, u) + ∂g

∂z(y, z, u)

·[

K (zset − z) + K

τi

∫ t

0

(zset − z

)dτ

]+

∂g

∂u(y, z, u) · du

dt= 0 (c)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

[12]

Generic Model Control for Battery Models

In this section, we discuss the performance of generic model con-trol strategy for different types of battery models.

Thin film nickel hydroxide electrode.—Consider the DAE modeldescribing a galvanostatic charge of a thin-film nickel hydroxide elec-trode as described by Wu et al.23 The model equations are given by:

d y1

dt= j1 · W

FρV[13]

j1 + j2 = Iapp [14]

d Q

dt= Iapp [15]

j1 = i0,1

[2 (1 − y1) exp

((φ1 − φ01) F

2RT

)−2y1 exp

(− (φ1 − φ01) F

2RT

)][16]

j2 = i0,2

[exp

((φ1 − φ02) F

RT

)− exp

(− (φ1 − φ02) F

RT

)][17]

where y1 is the mole fraction of nickel hydroxide, φ1 represents thepotential difference at the solid-liquid interface and Q is the total




Table I. Parameters of nickel hydroxide electrode.

Symbol Parameter Value Units

F Faraday constant 96,487 C/molR Gas constant 8.314 J/(mol K)T Temperature 303.15 Kφ01 Equilibrium potential 0.420 Vφ02 Equilibrium potential 0.303 VW Mass of active material 92.7 gV Volume 1 × 10−5 m3

i01 Exchange current density 1 × 10−4 A/cm3

i02 Exchange current density 1 × 10−10 A/cm3

P Density 3.4 g/cm3

charge stored in the battery. Iapp represents the applied current andis the manipulated variable for this model. The initial condition ofthe variables is y1(0) = 0.05, φ1(0) = 0.350236, Q(0) = 0 andIapp(0) = 1 × 10−5. The parameters for this model are given in TableI.

For this model, three different objectives are studied as discussedbelow.

a) In the first objective, we solve the model for mole-fraction (y1) toreach its desired set point (yset

1 ). Even though this model is a DAE,the equation for the total derivative of the measured variable ( d y1

dt )is directly available from the model (Equation 13). This can becompared with Equation 2 to realize Y = y1.

The GMC formulation for this case is given by:

d y1

dt= j1 · W

FρV[18]

d y1

dt= K · (

yset1 − y1 (t)

) + K

τir (t) [19]

where r (t) represents the integral of the offset and is defined in Equa-tion 20 as

dr

dt= (

yset1 − y1 (t)

)[20]

From Equations 18 and 19, the following relation can be derived:

j1W

FρV= K

(yset

1 − y1 (t)) + K

τir (t) [21]

Further, Equation 14 can be used to get:

(Iapp − j2

)W

FρV= K

(yset

1 − y1 (t)) + K

τir (t) [22]

Using Equation 22, an explicit relation for Iapp can be written as,

Iapp = FρV

W

(K

(yset

1 − y1(t)) + K

τir (t)

)+ j2 [23]

Equations 13–17 along with 23 are solved simultaneously to get theGMC control profile. The final set of equations (GMC model) to be

solved for this case are summarized again in 24.

d y1

dt= j1 · W

FρV(a)

j1 + j2 = Iapp (b)

d Q

dt= Iapp (c)

dr

dt= (

yset1 − y1 (t)

)(d)

Iapp = FρV

W

(K

(yset

1 − y1(t)) + K

τir (t)

)+ j2 (e)

j1 = i0,1

[2 (1 − y1) exp

((φ1 − φ01) F

2RT

)

−2y1 exp

(− (φ1 − φ01) F

2RT

)](f)

j2 = i0,2

[exp

((φ1 − φ02) F

RT

)− exp

(− (φ1 − φ02) F

RT

)](g)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭[24]

It should be noted that the manipulated variable (Iapp) is governedby Equation 24(e), whereas Equation 24(a), 24(b) and 24(c) gov-ern y1, φ1 and Q respectively. The initial condition for r (t) is as-sumed to be zero, the proportionality constant K is taken to be 0.02,and τi is assumed to be 3500. Figure 1a shows the profiles formole fraction (y1) for various set-points of yset

1 = [0.5, 0.6, 0.7].Figure 1b shows the profiles of manipulated variable (Iapp)thatdirects mole fraction (y1) to reach its desired set-point. An up-per bound of 120 μA/cm2 is used for this case. The strategyto implement bounded currents is explained in detail in the nextexample.

b) In the second case, the same DAE model is solved for totalcharged stored (Q) to reach its desired set point (Qset ). This caseis similar to the case (a) discussed above. Equation 15 can beused to get the explicit derivative of the measured variable Q.Hence, for this case:

Y = Q

The GMC formulation is given by the following set ofmodels,

d Q

dt= K

(Qset − Q (t)

) + K

τir (t) [25]

dr

dt= (

Qset − Q (t))

[26]

From Equations 15 and 25, we can get:

Iapp (t) = K(Qset − Q (t)

) + K · r (t)

τi[27]




Figure 1. Profiles of (a) Mole fraction (y1) and (b) Applied current (Iapp) with time in the thin film nickel hydroxide electrode section (a).

If there are no bounds on current, the final set of equations to be solvedfor this case is given as follows:

d y1

dt= j1 · W

FρV(a)

j1 + j2 = Iapp (b)

d Q

dt= Iapp (c)

dr

dt= (

Qset − Q(t))

(d)

Iapp = K(Qset − Q (t)

) + K

τir (t) (e)

j1 = i0,1

[2 (1 − y1) exp

((φ1 − φ01) F

2RT

)−2y1 exp

(− (φ1 − φ01) F

2RT

)](f)

j2 = i0,2

[exp

((φ1 − φ02) F

RT

)− exp

(− (φ1 − φ02) F

RT

)](g)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭[28]

The controller parameters for this case are K = 0.1 and τi = 20, andthe initial condition for r (t) is taken to be zero. Different set pointsfor charge stored are tried for this case.

In case if there are bounds on the applied current, it is bounded byapplying upper and lower limits using relation given in Equation 27.Here, we assume a lower bound of zero, and an upper bound of 120μA/cm2. Such a case can be solved by introducing a dummy variable(I1(t)), that represents the applied current in the GMC Equation 28(e),and the bounds are then applied on the current variable (I1(t)) as

shown in Equation 29.

Iapp(t) = min(12 × 10−5, max (0, I1 (t))

)[29]

For the bounded case, the final equations to be solved are summarizedbelow.

d y1

dt= j1 · W

FρV(a)

j1 + j2 = Iapp (b)

d Q

dt= Iapp (c)

dr

dt= (

Qset − Q(t))

(d)

I1 = K(Qset − Q (t)

) + K

τir (t) (e)

Iapp = min(12 × 10−5, max (0, I1)

)(f)

j1 = i0,1

[2 (1 − y1) exp

((φ1 − φ01) F

2RT

)−2y1 exp

(− (φ1 − φ01) F

2RT

)](g)

j2 = i0,2

[exp

((φ1 − φ02) F

RT

)− exp

(− (φ1 − φ02) F

RT

)](h)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭[30]

Figure 2a shows the profiles for charge stored with time for variousset points of Qset = [0.15, 0.2, 0.3] and Figure 2b shows the corre-sponding control profile current for different set points. It can be seenthat the charge stored reaches its set point in all the cases.

c) In the third objective, the DAE model is solved for potential ofthe battery, φ1 (which is an algebraic variable in this model) to




Figure 2. Profles of (a) Charge stored (Q) and (b) Applied current (Iapp) with time in the thin film nickel hydroxide electrode section (b).

reach its desired set point φset1 . For this case,

Y = φ1

The full derivative ( dφ1dt ) is not given by the model equations and

∂g∂u (y, z, u) (in Equation 9) is not zero. The GMC formulation forthis case is given by equations described earlier for the DAE model.Equations 16 and 17 can be substituted in 14 to get:

Iapp =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

i0,1

[2 (1 − y1) exp

((φ1 − φ01) F

2RT

)−2y1 exp

(− (φ1 − φ01) F

2RT

)]+

i0,2

[exp

((φ1 − φ02) F

RT

)− exp

(− (φ1 − φ02) F

RT

)]

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭[31]

Equation 31 can now be differentiated to get:

d Iapp

dt

=

⎡⎢⎢⎣i0,1

⎡⎣2 (1 − y1) exp (a1) · F2RT · dφ1

dt + 2 exp (a1)(− d y1

dt

)−2y1 exp (−a1) · (− F

2RT

) · dφ1dt − 2 exp (−a1)

( d y1dt

)⎤⎦ +

i0,2

[exp (a2) · F

RT · dφ1dt − exp (−a2) · (− F

RT

) · dφ1dt

]⎤⎥⎥⎦

[32]

a1 = (φ1 − φ01) F

2RT, a2 = (φ1 − φ02) F

2RT[33]

Equation 32 is rewritten for dφ1dt to get:

dφ1

dt=

d Iapp

dt + 2i0,1 exp (a1)( d y1

dt

) + 2i0,1 exp (−a1)( d y1

dt

)⎡⎣i0,1

[2 (1 − y1) exp (a1) · F

2RT − 2y1 exp (−a1) · (− F2RT

)] +

i0,2

[exp (a2) · F

RT − exp (−a2) · (− FRT

)]⎤⎦

[34]Also, the equations for the GMC formulation is added as:

dφ1

dt= K

(φset

1 − φ1 (t)) + K · r (t)

τi[35]

dr

dt= (

φset1 − φ1 (t)

)[36]


d Iapp

dt + 2i0,1 exp (a1)( d y1

dt

) + 2i0,1 exp (−a1)( d y1

dt

)⎡⎢⎢⎢⎢⎢⎢⎢⎣i0,1

⎡⎢⎢⎢⎣2 (1 − y1) exp (a1) · F

2RT−

2y1 exp (−a1) ·(

− F

2RT

)⎤⎥⎥⎥⎦+

i0,2

[exp (a2) · F

RT− exp (−a2) ·

(− F

RT

)]

⎤⎥⎥⎥⎥⎥⎥⎥⎦

= K(φset

1 − φ1 (t)) + K · r (t)

τi[37]

Equations 13–17 are solved together along with Equations 36 – 37 toobtain the control profiles for this case. In order to implement boundson the applied current, a similar dummy variable (I1(t)), is added inthe GMC equation. The final set of model equations solved for this




Figure 3. Profiles of (a) Potential(φ1) and (b) Applied current (Iapp) with time in the thin film nickel hydroxide electrode section (c).

case are summarized in Equation 38.

d y1

dt= j1 · W

FρV(a)

j1 + j2 = Iapp (b)

d Q

dt= Iapp (c)

dr

dt= (

φset1 − φ1(t)

)(d)

d I1

dt+ 2i0,1 exp (a1)

(d y1

dt

)+ 2i0,1 exp (−a1)

(d y1

dt

)⎡⎢⎢⎢⎣

i0,1

[2 (1 − y1) exp (a1) · F

2RT− 2y1 exp (−a1) ·

(− F

2RT

)]+

i0,2

[exp (a2) · F

RT− exp (−a2) ·

(− F

RT

)]⎤⎥⎥⎥⎦

= K(φset

1 − φ1 (t)) + K · r (t)

τi(e)

Iapp = min(2 × 10−5, max (0, I1)

)(f)

j1 = i0,1

[2 (1 − y1) exp

((φ1 − φ01) F

2RT

)−2y1 exp

(− (φ1 − φ01) F

2RT

)](g)

j2 = i0,2

[exp

((φ1 − φ02) F

RT

)− exp

(− (φ1 − φ02) F

RT

)](h)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭[38]

It can be noted that in this case, Equation 38b determines the variationof the measured variable with time, and Equations 38e and 38f governthe control profile. The controller parameters for this case are K =

0.0250 and τi = 15. Figure 3 shows the results obtained for this case.Figures 3a and 3b show the potential and current profiles respectivelyobtained for different set points φset

1= [0.4, 0.5, 0.6].

Single particle model.—A single particle model10 for an electrodeis used to illustrate the concept for a model with multiple algebraicequations in the DAE system. Consider diffusion of a particle insidea spherical electrode. The model equations for this case are given by:

∂c

∂t= 1

r 2·(

∂

∂r

(r 2 ∂c

∂r

))[39]

With the boundary conditions

At r = 0,∂c

∂r= 0 [40]

and at r = 1,∂c

∂r= jp [41]

where c is the scaled concentration of the particle, r is the scaled radialvariable in the spherical coordinate, and jp is the scaled flux of theparticle inside the electrode, which is a function of the applied currentdensity.

The surface concentration (cs) is given by:

cs = c (r )|r=1 [42]

In order to solve the model, a finite difference solution is used. Thedetails of implementing finite difference solution (method of linesapproach) can be found elsewhere,24 and is not discussed here. Thefinal set of equations obtained after implementing finite difference forN = 19 internal node points are given by:

dCi

dt= Ci+1 − 2Ci + Ci−1

h2+ 2

i · h

Ci+1 − Ci−1

2 · h, i = 1 . . . 19

[43]

C1 − C0

h= 0 [44]




Table II. Expression for the open circuit potential in the single particle model.

E0 =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

a1 + a2Cs + a3Cs2 + a4Cs

3+

a5(Cs − a)2(1

2tanh(1000(Cs + 1)) + tanh(1000(Cs − a)))+

a6(Cs − b)3(1 − 1

2tanh(1000(Cs + 1)) + tanh(1000(Cs − b)))+

a7(Cs − c)3(1 − 1

2tanh(1000(Cs + 1)) + tanh(1000(Cs − c)))+

a8(Cs − d)3(1 − 1

2tanh(1000(Cs + 1)) + tanh(1000(Cs − d)))+

a9(Cs − e)3(1 − 1

2tanh(1000(Cs + 1)) + tanh(1000(Cs − e)));

a1 = 9.41894182550,

a2 = −20.8089294802,

a3 = 26.0321098756,

a4 = −11.2415464828,

a5 = −292.586540767,

a6 = −.212804379260

a7 = 125.437252178,

a8 = −198.413506247,

a9 = 164.029238273

a = 0.96, b = 0.733,

c = 0.575, d = 0.535,

e = 0.466

CN+1 − CN

h= jp [45]

Cs = cs = CN+1 [46]

h = 1

N + 1[47]

where Ci is the concentration of the particle at the ith node point inthe radial direction. The manipulated variable for this case, is thedimensionless flux ( jp). The potential of the electrode (V ) is definedaccording to a Nernst equation, as given by:

V (t) = E0 − RT

Flog (Cs (t)) [48]

where E0 is the open-circuit potential of the electrode and is assumedto be a function of the surface concentration (Cs) of the electrode (asis the case in most battery models), R is the ideal gas constant. F isthe Faraday’s constant, and T is the temperature of the electrode, andis assumed to be 298 K. The expression for E0 used as a functionof the surface concentration is shown in Table II. The values of pa-rameters are assumed only for theoretical purposes, and not versusany standard electrode. The initial conditions of all the variables areCi (0) = 0.5,V (0) = 4.09 and jp (0) = 0.

The measured variable is assumed to be V, and a set points ofV set = [3.2V, 3.5V ] are taken.

As in the previous case, the derivative of potential ( dVdt ) is calculated

by differentiating Equation 48. For this case,

Y = VdVdt = d E0

dCs· dCs

dt − RTF · 1

Cs· dCs

dt[49]

Moreover, ∂g∂u (y, z, u) (Equation 9) is not zero for this case.

The GMC formulation is written as follows:

dV

dt= K

(V set − V (t)

) + K

τir (t) [50]

dr

dt= (

V set − V (t))

[51]


d E0

dCs· dCs

dt− RT

F· 1

Cs· dCs

dt= K

(V set − V (t)

)+ K

τir (t) [52]

Equation 45 can be differentiated to get the derivative for dCsdt (after

substituting Equation 46) as

dCs

dt= dCN+1

dt= h · d jp

dt+ dCN

dt[53]

From Equations 52 and 53, we get:

d E0

dCs·(

h · d jp

dt+ dCN

dt

)− RT

F· 1

Cs·(

h · d jp

dt+ dCN

dt

)= K

(V set − V (t)

) + K

τir (t) [54]

Equation 54 is solved along with Equations 43–48 to obtain the GMCcontrol law. When there is no bound on the scaled flux, the equationsto be solved are summarized as:

dCi

dt= Ci+1 − 2Ci + Ci−1

h2+ 2

i · h

Ci+1 − Ci−1

2 · h, i = 1 . . . 19(a)

C1 − C0

h= 0 (b)

CN+1 − CN

h= jp (c)

Cs = cs = CN+1 (d)

Cavg (t) =C0 (t) + 2

(∑Ni=1 Ci (t)

)+ CN+1 (t)

2N + 2(e)

h = 1

N + 1(f)

V (t) = E0 − RT

Flog (Cs (t)) (g)

dr

dt= (

V set − V (t))

(h)

d E0

dCs·(

h · d jp

dt+ dCN

dt

)− RT

F· 1

Cs·(

h · d jp

dt+ dCN

dt

)= K

(V set − V (t)

) + K

τir (t) (i)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭[55]

The controller tuning parameter K is chosen as 1 and τi is assumed tobe 1000. Figure 4a shows the profiles of potential with time for twodifferent set points and the corresponding control profiles are shown in




Figure 4. Profiles of (a) potential (b) scaled flux (jp) and (c) surface concentration (Cs) with time for the single particle model.

Figure 4b. The profiles for surface concentration are shown in Figure4c.

In order to implement bounds, same procedure as discussed for theunbounded case is followed, and the equations are again summarizedbelow but with bounds on the manipulated variable jp .

dCi

dt= Ci+1 − 2Ci + Ci−1

h2+ 2

i · h

Ci+1 − Ci−1

2 · h, i = 1 . . . 19 (a)

C1 − C0

h= 0 (b)

CN+1 − CN

h= jp (c)

Cs = cs = CN+1 (d)

h = 1

N + 1(e)

V (t) = E0 − RT

Flog (Cs (t)) (f)

dr

dt= (

V set − V (t))

(g)

d E0

dCs·(

h · d j1dt

+ dCN

dt

)− RT

F· 1

Cs·(

h · d j1dt

+ dCN

dt

)(h)

= K(V set − V (t)

) + K

τir (t) (i)

jp = min (0.20, max (0, j1)) (j)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭[56]

Figure 5 shows the profiles obtained for an upper bound on jp of 0.2for different set points on voltage. The control profiles obtained in thiscase again drive the measured variable to its required set point.

Reformulated pseudo 2 dimensional (P2D) model.—Pseudo 2-Dimensional (P2D) model proposed by Doyle et al.25 is one ofthe most popular physics-based lithium-ion battery model. Otherapproaches26,27 have been proposed for efficient simulation of bat-

tery models. Northrop et al.28,29 published a reformulated batterymodel, based on the original P2D model, and the reformulated modelwas used in this work to obtain the control profiles after implement-ing GMC. The set of equations and the corresponding parametersof the battery model are listed in Table III, Table IV and Table V(Reference 29).

We present two cases for this model, even though GMC formula-tion can be applied for more variables.

Set point for the charge stored in the battery.—In this case, theobjective is to find the current profiles to obtain the final charge storedto reach a desired set pointQset , with a bound on the applied current.Mathematically,

Find Iapp(t) subject to the model equations given in Table III, andthe constraints

Q(t f

) = Qset [57]

0 ≤ Iapp(t) ≤ 3C [58]

where Iapp(t) is the applied current, Q(t) is the charged stored in thebattery, Qset is the set-point for the charge, C is the capacity of thebattery and t f is the final time. The value of Qset for this study is takento be 14 A-hr.

For this problem (comparing with Equation 2),

Y = Q

The full derivative of the measured variable is directly available fromthe model thereby resulting in a direct equation for Iapp instead ofd Iapp

dt . For this case, in Equation 9, ∂g∂u (y, z, u) is not zero.

The equation for charge stored is given by:

d Q

dt= Iapp [59]

As in the previous examples, the GMC equation can be written as:

d Q

dt= K

(Qset − Q (t)

) + K

τir (t) [60]




Table III. Equations for P2D thermal model.Governing Equation Boundary ConditionsPositive Electrode

εp∂cp∂t = 1

l p

∂∂ X [

Deff,pl p

∂cp∂ X ] + ap(1 − t+) jp

∂cp

∂ X|X=0= 0

−Deff,p

l p

∂cp

∂ X|X=1 = −Deff,s

ls

∂cs

∂ X|X=0

−σeff,pl p

(∂�1,p∂ X ) − κeff,p

l p(∂�2,p∂ X ) + 2κeff,p RT

F(1−t+)

l p( ∂ ln cp

∂ X ) = I

∂�2,p

∂ X|X=0= 0

−κeff,p

l p

∂�2,p

∂ X|X=1 = −κeff,s

ls

∂�2,s

∂ X|X=0

1l p

∂∂ X [

σeff,pl p

∂∂ X �1,p] = ap F jp

(1

l p

∂�1,p

∂ X)|X=0 = − I

σe f f,p∂�1,p

∂ X|X=1= 0

∂csp

∂t = 1r2

∂∂r [r2 Ds

p∂cs

p∂r ]

∂csp

∂r|r=0 = 0

−Dsp

∂csp

∂r|r=Rs = jp

ρpC p,pdTpdt = 1

l p

∂∂ X [ λp

l p

∂Tp∂ X ] + Qrxn,p + Qrev,p + Qohm,p

−K ef f p∂Tp

∂ X|X=0 = henv(Tp|X=0 − Tair )

−λp

l p

∂Tp

∂ X|X=1 = −λs

ls

∂Ts

∂ X|X=0

Separator

εs∂cs∂t = 1

ls∂

∂ X [ Deff,sls

∂cs∂ X ]

cp |X=1 = cs |X=0

cs |X=1 = cn |X=0

− κeff,sls

( ∂�2,s∂ X ) + 2κeff,s RT

F(1−t+)

ls( ∂ ln cS

∂ X ) = I�2,p |X=1 = �2,s |X=0

�2,s |X=1 = �2,n |X=0

ρsC p,sdTsdt = 1

ls∂

∂ X [ λsls

∂Ts∂ X ] + Qohm,s

Tp |X=1 = Ts |X=0

Ts |X=0 = Tn |X=1

Negative Electrode

εn∂cn∂t = 1

ln∂

∂ X [ Deff,nln

∂cn∂ X ] + an(1 − t+) jn

∂cn

∂ X|X=1= 0

−Deff,s

ls

∂cs

∂ X|X=1 = −Deff,n

ln

∂cn

∂ X|X=0

−σeff,nln

( ∂�1,n∂ X ) − κeff,n

ln( ∂�2,n

∂ X ) + 2κeff,n RTF

(1−t+)ln

( ∂ ln cn∂ X ) = I

�2,n |X=1= 0

−κeff,s

ls

∂�2,s

∂ X|X=1 = −κeff,n

ln

∂�2,n

∂ X|X=0

1ln

∂∂ X [ σeff,n

ln∂

∂ X �1,n] = an F( jn + jsei )

∂�1,n

∂ X|X=0= 0

(1

ln

∂�1,n

∂ X)|X=1 = − I

σe f f,n

∂csn

∂t = 1r2

∂∂r [r2 Ds

n∂cs

n∂r ]

∂csn

∂r|r=0 = 0

−Dsn∂cs

n

∂r|r=Rs = jn

ρnC p,ndTndt = 1

ln∂

∂ X [ λnln

∂Tn∂ X ] + Qrxn,n + Qrev,n + Qohm,n

−λs

ls

∂Ts

∂ X|X=1 = −λn

ln

∂Tn

∂ X|X=0

−K ef fn∂Tn

∂ X|X=1 = henv(Tair − Tn |X=1)

dr

dt= Qset − Q (t) [61]

From Equations 59 and 60, the equation for applied current can bewritten as:

Iapp = K(Qset − Q (t)

) + K

τir (t) [62]

The bounds in Equation 58, are applied on the current as:

Iapp(t) = min

(54, max

(0, K

(Qset − Q (t)

) + K

τir (t)

))[63]

where 54 A/m2 is assumed to be the current equivalent for 3 C. Thecontroller tuning parameter is chosen as K = 0.01, and τi = 1e − 4.Equation 63 can be used along with the model equations mentionedin Table III, to obtain the GMC control profile.




Figure 5. Profiles of (a) Potential, (b) Scaled flux (jp)and (c) Surface concentration (Cs) with time for the sin-gle particle model.

Figure 6a, shows the profile of charge stored reaching the desiredset-point and the corresponding applied current profile is shown inFigure 6b. Since for this case, there is no constraint given on tempera-ture, or voltage, the battery is charged at a constant current of 3C thatis the maximum bound on the current, until the charge stored reachesthe desired set-pointQset . After t = 934 seconds, the current falls tozero in order to keep the charge stored at the desired set-pointQset .Figures 6c and 6d show the corresponding changes in voltage andtemperature of this constant current charging case, respectively. Thetemperature and voltage both increase steadily as expected until thecurrent is non-zero. After t = 934 seconds, when the current becomeszero, the temperature starts to fall, whereas voltage remains constantat the open-circuit value.

Set point for voltage of the cell.—Consider a cell being chargedfrom a fully discharged state. The set point for the voltage whilecharging is set to be V set = 4.2 V. In the P2D model, the voltage of

the cell (V ) is given by the equation:

V (t) = φ1 (t)|x=0 − φ1 (t)|x=L pos+Lsep+Lneg[64]

where φ1(t) denotes the potential of the solid phase, and L pos , Lsep ,Lneg , represent the thicknesses of the positive electrode, separator andnegative electrode of the cell, respectively.

This case is similar to the case presented in Thin film nickel hy-droxide electrode section (c), where the explicit derivative for themeasured variable (voltage) is not given by the model equations.

For this case,

Y = V (t)

and ∂g∂u (y, z, u) (in Equation 9) is not zero.

Through substitution of other model equations, it can be shown thatthe potential of the cell is directly dependent on the applied current(Iapp). Equation 64 is differentiated with time to get the derivativeof potential. In order to find derivative for other terms occurring in

Figure 6. Profiles of (a) Charge stored, (b) Applied current, (c)Potential and (d) Temperature with time for the reformulated P2Dmodel.




Table IV. Additional Equations.29

Qrxn,i = Fai ji (�1,i − �2,i − Ui ), i = p, n

Qrev,i = Fai ji Ti∂Ui∂T , i = p, n

Qohm,i = σeff,i ( 1li

∂�1,i∂ X )2 + κeff,i ( 1

li

∂�2,i∂ X )2

+ 2κeff,i RTiF (1 − t0+) 1

l2i

1ci

∂ci∂ X

∂�2,i∂ X , i = p, n

Qohm,s = κeff,s ( 1ls

∂�2,s∂x )2 + 2κeff,s RTs

F (1 − t0+) 1cs

1l2s

∂cs∂ X

∂�2,i∂ X

Ui (Ti , θi ) = Ui,ref (Tref , θi ) + (Ti − Tref )[dUidT ]|Tref , i = p, n

Dsi,e f f = Ds

i exp(− EDs

iaR [ 1

T − 1Tre f

]), i = p, n

ki,e f f = ki exp(− EkiaR [ 1

T − 1Tre f

]), i = p, n

Equation 64, all the algebraic equations are differentiated with time,to convert the system of differential algebraic equations (DAEs) to asystem of ordinary differential equations (ODEs). GMC formulationis then applied for this case as:

dV

dt= K (4.2 − V (t)) + K

τir (t) [65]

dr

dt= 4.2 − V (t) [66]

The dVdt term appearing in the differentiation of all the algebraic

equations, is eliminated by substituting Equation 65. The math fol-lowed is similar to the set point for potential explained for potentialexplained in the thin film nickel hydroxide section (c). The tuningparameter K is assumed to be 0.01 and τi is taken to be 100. Figure 7(solid red curve) shows the results when no bound is imposed onthe applied current and the potential for the PI case. In this case, thevoltage of the battery goes above 4.2 V, ultimately driving the voltageto its desired set point. However, for these sets of tuning parame-ters, the battery is overcharged before set-point for voltage is reached,which causes an increase in the rate of battery degradation. Figure 7also shows the effect of increasing the τi value to 1e9 (dashed greencurve), which drives the voltage to its set-point without going above

the set-point. However, it should be noted that increasing the τi valuedecreases the impact of the integral term.

Hence, for this case, GMC formulation is applied using only aproportional term (P-Controller), as:

dV

dt= K (4.2 − V (t)) [67]

For this case, the voltage of the battery reaches its set point of 4.2 V,whereas the current increases first until 189 s, and decreases thereafter,as shown in Figure 8.

Set point for voltage with bounds on applied current.—In the nextcase, the set point of 4.2 V is applied to the potential as in case (b),but additional bounds are added for applied current as shown below.

0 ≤ Iapp (t) ≤ 30A/m2 [68]

This case is similar to the single particle model case described above.Figure 9a shows the change in voltage vs time for this case, and Figure9b shows the GMC control profile for current.

Through these examples, it can be seen that the voltage reaches itsdesired set point for both, the bounded as well as the unbounded cur-rent case. The bounded case is closer to the real life scenario, where anupper limit is usually applied on the charging current. The advantageof implementing control through GMC is that the variables are onlyto be integrated in time. We implemented the control scheme usingthe direct iteration-free DAE numerical solvers previously reported byLawder et al.30 This makes the approach more robust and fail-proof,as standard optimizers fail to find the optimal solution often timesbecause of their inability to find consistent initial conditions for alge-braic variables. It should be noted that for simple first order or secondorder processes, the time constants of the processes can be related tothe tuning of the GMC parameters. However, physics-based batterymodels have multiple dynamics, and we choose an approach to modifythe tuning parameters until no effect is seen in the profiles of controland measured variables. The arbitrariness in the tuning parameters isan inherent limitation of the GMC approach.

Performance of GMC under Model Uncertainty

Lee and Sullivan14 have discussed the impact of uncertainty inparameters in their original manuscript. In this section, we summarizethe same and discuss the uncertainty for the thin-film electrode model.

Table V. List of parameters for the P2D model.29

Symbol Parameter Positive Electrode Separator Negative Electrode Units

σi Solid phase conductivity 100 100 S/mε f,i Filler fraction 0.025 0.0326εi Porosity 0.385 0.724 0.485Brugg Bruggman Coefficient 4D Electrolyte diffusivity 7.5 × 10−10 7.5 × 10−10 7.5 × 10−10 m2/sDs

i Solid Phase Diffusivity 1.0 × 10−14 3.9 × 10−14 m2/ski Reaction Rate constant 2.334 × 10−11 5.031 × 10−11 mol/(s m2)/(mol/m3)1+αa,i

csi,max Maximum solid phase concentration 51554 30555 mol/m3

csi,0 Initial solid phase concentration 25751 26128 mol/m3

c0 Initial electrolyte concentration 1000 mol/m3

Rp,i Particle Radius 2.0×10−6 2.0×10−6 Mai Particle Surface Area to Volume 885000 723600 m2/m3

li Region thickness 80×10−6 25×10−6 88×10−6 Mt+ Transference number 0.364F Faraday’s Constant 96487 C/molR Gas Constant 8.314 J/(mol K)Tre f Temperature 298.15 KP Density 2500 1100 2500 kg/m3

Cp Specific Heat 700 700 700 J/(kg K)� Thermal Conductivity 2.1 0.16 1.7 J/(m K)

EDs

ia Activation Energy for Temperature Dependent Solid Phase Diffusion 5000 5000 J/mol

Ekia Activation Energy for Temperature Dependent Reaction Constant 5000 5000 J/mol




Figure 7. Profiles of (a) Potential (b) Applied current vs timefor the P2D model obtained using PI controller.

Consider the plant model given by Equations 7 and 8 and theprocess model containing uncertainty given by:

dy

dt= f (y, z, u) [69]

g (y, z, u) = 0 [70]

The GMC formulation for the process model, where set point is givenfor the algebraic variable is written as:

dg

dt= ∂ g

∂y(y, z, u) · f (y, z, u) + ∂ g

∂y(y, z, u)

·[

K(zset − z

) + K

τi

∫ t

0

(zset − z

)dτ

]+ ∂ g

∂u(y, z, u) · du

dt= 0

[71]

To solve GMC under model uncertainty, Equation 71 is used to obtainthe control profile and solved along with Equations 7 and 8. Forthe thin-film electrode model, the model uncertainty is consideredby perturbing the parameters i0,1, φ01, T , W by ±10%. Potential(algebraic variable) is chosen to reach its desired set-point φset

1 inthe presence of uncertain parameters mentioned above. The potentialprofiles obtained are shown in Figure 10 and it can be observed that theGMC approach gives robust performance under model uncertainty, asthe controlled variable reaches its desired set-point in every case.

Conclusions

Generic model control is applied for different types of batterymodels such as thin film nickel hydroxide electrode, single particlemodel, along with a reformulated pseudo 2D model. The conven-tional GMC approach is extended for DAEs as well as for problems

Figure 8. Profiles of (a) Potential (b) Applied current density vs time for the reformulated P2D model using P-controller.




Figure 9. Profiles of (a) Potential (b) Applied current density vs time for the reformulated P2D model using P-Controller.

involving the direct correlation between the measured and the ma-nipulated variable. The results are presented for various case studies,and the measured variable is shown to reach the desired set pointin all cases. The approach is found to be extremely robust and fail-safe. Future work involves the implementation of the GMC controlscheme for problems involving constraints on various state variables(for example, bounds on voltage and temperature for the P2D model).In the GMC framework, problems involving constraints or bounds

has to be solved as an optimization problem. The constraint handlingstrategy and the need for performing optimization in the GMC frame-work will be explored in a future publication. The method will alsobe extended to explore the control profiles for single input multipleobjective case, like minimizing the capacity fade while maximizingthe charge stored in the battery, while maintaining the temperatureand voltage constraints, and for process and plant models with timedelays.

Figure 10. Profiles of potential reaching its sepecificed set-point, where parameters (a) Exchange current density, (b) Equilibrium potential, (c) Temperature and(d) Mass of the active material are perturbed by ±10%.




Acknowledgments

The authors thank the Department of Energy (DOE) for providingpartial financial support for this work, through the Advanced ResearchProjects Agency (ARPA-E) award number DE-AR0000275. The au-thors would also like to thank the Clean Energy Institute (CEI) atthe University of Washington (UW) and the Washington ResearchFoundation (WRF) for their monetary support for this work.

List of Symbols

c Electrolyte concentrationcs Solid Phase ConcentrationD Liquid phase Diffusion coefficientDef f Effective Diffusion coefficientDs Solid phase diffusion coefficientEa Activation EnergyF Faraday’s ConstantI app Applied Currentj Pore wall fluxK Proportional Gaink Reaction rate constantl Length of regionR Particle Radius, or Residualt+ Transference numberT TemperatureU Open Circuit PotentialW Mass of the active material

Greek

ε Porosityε f Filling fractionθ State of Chargeκ Liquid phase conductivityσ Solid Phase Conductivity�1 Solid Phase Potential�2 Liquid Phase Potentialτi Integral Time Constant

Subscripts

e f f Effective, as for diffusivity or conductivityc Related to Electrolyte concentrationcs Related to Solid Phase concentrationn Related to the negative electrode—the anodep Related to the positive electrode—the cathodes Related to the separator�1 Related to the solid phase potential�2 Related to the liquid phase potential

Superscripts

avg Average, as for solid phase concentrationsur f Surface, as for solid phase concentrations Related to Solid Phase

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